Differential Equations and Linear Algebra

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This very accessible guide offers a thorough introduction to the basics of differential equations and linear algebra. Expertly integrating the two topics, it explains concepts clearly and logically -without sacrificing level or rigor - and supports material with a vast array of problems of varying levels for readers to choose from. Promotes in-depth understanding (vs. rote memorization) - enabling readers to fully comprehend abstract concepts and finish with a solid and working knowledge of linear mathematics. Offers one of the most lucid and clearly written narratives on the subject, with material that is accessible to the average reader, yet challenging to all. Presents a greater emphasis on geometry to help users better visualize the abstract concepts, and illustrates all concepts with an ample amount of worked examples. Second Edition highlights include new discussions direction fields and Euler's method for first order differential equations; row space and column space of a matrix, and the rank-nullity theorem; non-linear systems of differential equations, including phase plane analysis; and change of variables for differential equations. Now features a chapter on second order linear differential equations that isnot based on vector space methods to gives users a firmer grasp of the differential equation concept early on, and also on the solution techniques for this important class of differential equations.

Preface

xiii

First-Order Differential Equations

1

(105)

How Differential Equations Arise

1

(8)

Basic Ideas and Terminology

9

(9)

The Geometry of First-Order DE

18

(11)

Separable DE

29

(9)

Some Simple Population Models

38

(6)

First-Order Linear DE

44

(7)

Two Modeling Problems Governed by First-Order Linear DE

51

(10)

Change of Variables

61

(10)

Exact DE

71

(9)

Summary of Techniques for Solving First-Order DE

80

(2)

Numerical Solution to First-Order DE

82

(9)

Some Higher Order DE

91

(5)

The Phase Plane

96

(10)

Second-Order Linear Differential Equations

106

(61)

Basic Theoretical Results

107

(7)

Reduction of Order

114

(5)

Second-Order Homogeneous Constant Coefficient Linear DE

119

(6)

The Method of Undetermined Coefficients

125

(8)

Complex-Valued Trial Solutions

133

(3)

Oscillations of a Mechanical System

136

(14)

RLC Circuits

150

(4)

The Variation-of-Parameters Method

154

(6)

A DE with Nonconstant Coefficients

160

(7)

Matrices and Systems of Linear Algebraic Equations

167

(65)

Matrices: Definitions and Notation

168

(4)

Matrix Algebra

172

(13)

Terminology and Notation for Systems of Linear Equations

185

(5)

Elementary Row Operations and Row-Echelon Matrices

190

(11)

Gaussian Elimination

201

(11)

The Inverse of a Square Matrix

212

(10)

Elementary Matrices and the LU Factorization

222

(10)

Determinants

232

(39)

The Definition of a Determinant

232

(10)

Properties of Determinants

242

(11)

Cofactor Expansions

253

(11)

Summary of Determinants

264

(7)

Vector Spaces

271

(84)

Vectors in Rn

272

(6)

Definition of a Vector Space

278

(7)

Subspaces

285

(7)

Spanning Sets

292

(8)

Linear Dependence and Linear Independence

300

(12)

Bases and Dimension

312

(12)

Row Space and Column Space

324

(5)

The Rank-Nullity Theorem

329

(5)

Inner Product Spaces

334

(9)

Orthogonal Sets of Vectors and the Gram-Schmidt Procedure

343

(8)

Summary of Results

351

(4)

Linear Transformations and the Eigenvalue/Eigenvector Problem

355

(67)

Definition of a Linear Transformation

356

(8)

Transformations of R2

364

(7)

The Kernel and Range of a Linear Transformation

371

(7)

Further Properties of Linear Transformations

378

(7)

The Algebraic Eigenvalue/Eigenvector Problem

385

(11)

General Results for Eigenvalues and Eigenvectors

396

(7)

Diagonalization

403

(7)

Orthogonal Diagonalization and Quadratic Forms

410

(10)

Summary of Results

420

(2)

Linear Differential Equations of Order n

422

(26)

General Theory for Linear Differential Equations

423

(7)

Constant Coefficient Homogeneous Linear DE

430

(5)

The Method of Undetermined Coefficients: Annihilators

435

(7)

The Variation-of-Parameters Method

442

(6)

Systems of Differential Equations

448

(85)

Introduction

448

(2)

First-Order Linear Systems

450

(5)

Vector Formulation

455

(6)

General Results for First-Order Linear Differential Systems

461

(5)

Homogeneous Constant Coefficient VDE: Nondefective Coefficient Matrix

466

(9)

Homogeneous Constant Coefficient VDE: Defective Coefficient Matrix

475

(10)

Variation-of-Parameters for Linear Systems

485

(5)

Some Applications of Linear Systems of Differential Equations

490

(11)

An Introduction to the Matrix Exponential Function

501

(5)

The Matrix Exponential Function and Systems of Differential Equations

506

(8)

The Phase Plane for Linear Autonomous Systems

514

(9)

Nonlinear Systems

523

(10)

The Laplace Transform and Some Elementary Applications

533

(46)

The Definition of the Laplace Transform

533

(6)

The Existence of the Laplace Transform and the Inverse Transform

539

(5)

Periodic Functions and the Laplace Transform

544

(3)

The Transform of Derivatives and the Solution of Initial-Value Problems